# Simple, Compound, APR, APY

## Simple vs Compound Interest

* Simple interest takes into account the principal, interest rate and period, but **ignores the effect of accumulated interest**.
* Compound interest accounts for accumulated interest: each period interest is generated on both the principal and the interest generated thus far up till the previous period.

{% hint style="info" %}
In each interest gathering period:

* simple: principal stays the same
* compound: principal grows
{% endhint %}

### Example

* Principal = $100
* annual interest rate = 5%
* period = 5 years

<figure><img src="https://1829638638-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F0k7YXwFGMFZcsyqkM4q1%2Fuploads%2FZAK4nA4Hee0Zkdnv03UX%2Fimage.png?alt=media&#x26;token=363f34cd-8343-4e29-875c-c6bd7b69a060" alt=""><figcaption></figcaption></figure>

{% hint style="info" %}
Compounding generates greater interest versus simple, for the same parameters.
{% endhint %}

## APR & APY

### APR

* APR represents the cost of borrowing or the interest rate on a loan without considering compounding.
* Uses simple interest in its calculation, ignoring any compounding effects.
* **`APR = Periodic Interest Rate * Number of Periods in Year`**
* APR will always be **equal to or lesser than** APY on a like-for-like basis in terms of interest rates.

{% hint style="info" %}
APR is typically used to compare different loan options or credit products. By using APR in their advertising, banks can make their loan products appear more attractive to potential borrowers because the APR does not reflect the impact of compounding over time.
{% endhint %}

### APY

$$APY  = (1 + \frac{r}{n})^n - 1$$, where r is the periodic interest rate, and n is the no. of periods.

* APY reflects the actual annualized return or cost on an investment or debt position, taking into account the compounding effect.
* The value of the position grows faster because every time the interest is calculated on the new balance, which includes the previous balance and the previous interest earned.

{% hint style="info" %}
- When comparing interest rates on a like-for-like basis, if both APR and APY are provided, the **APY will always be equal to or higher** than the APR due to the inclusion of compounding.&#x20;
- However, banks typically advertise loan products using APR, as it makes the interest rate appear lower and more enticing to borrowers.
{% endhint %}

{% hint style="success" %}
**Why have both APY and APY?**

* convenient depending on context to use one vs the other
*

    <figure><img src="broken-reference" alt=""><figcaption><p><a href="https://www.youtube.com/watch?v=87FAZSwmjCI">https://www.youtube.com/watch?v=87FAZSwmjCI</a></p></figcaption></figure>
{% endhint %}

### Converting APR to APY

* [https://seekingalpha.com/article/4519764-apr-vs-apy](https://seekingalpha.com/article/4519764-apr-vs-apy)
* [https://www.youtube.com/watch?v=eHNMr8TJArQ](https://www.youtube.com/watch?v=eHNMr8TJArQ)
